Complexity of Approximating the Vertex Centroid of a Polyhedron

Let P be an H-polytope in R with vertex set V . The vertex centroid is defined as the average of the vertices in V . We first prove that computing the vertex centroid of an H-polytope, or even just checking whether it lies in a given halfspace, are #P-hard. We also consider the problem of approximating the vertex centroid by finding a point within an ǫ distance from it and prove this problem to be #P-easy by showing that given an oracle for counting the number of vertices of an Hpolytope, one can approximate the vertex centroid in polynomial time. We also show that any algorithm approximating the vertex centroid to any “sufficiently” non-trivial (for example constant) distance, can be used to construct a fully polynomial-time approximation scheme for approximating the centroid and also an output-sensitive polynomial algorithm for the Vertex Enumeration problem. Finally, we show that for unbounded polyhedra the vertex centroid can not be approximated to a distance of d 1 2 −δ for any fixed constant δ > 0.